Domination spaces and factorization of linear and multilinear summing operators

TL;DR

This paper explores the factorization properties of various classes of linear and multilinear operators defined by summability conditions, unifying them under a comprehensive theoretical framework.

Contribution

It introduces a unified theory for factorization schemes applicable to a broad range of summing operators, including linear and multilinear cases, extending Pietsch type theorems.

Findings

Unified framework for summing operator factorizations

Includes linear and multilinear operator classes

Extends Pietsch type factorization results

Abstract

It is well known that not every summability property for non linear operators leads to a factorization theorem. In this paper we undertake a detailed study of factorization schemes for summing linear and nonlinear operators. Our aim is to integrate under the same theory a wide family of classes of mappings for which a Pietsch type factorization theorem holds. We analyze the class of linear operators that are defined by a summability inequality involving a homogeneous map. Our construction includes the cases of absolutely $p$-summing linear operators, $(p,\sigma)$-absolutely continuous linear operators, factorable strongly $p$-summing multilinear operators, $(p_1,\ldots,p_n)$-dominated multilinear operators and dominated $(p_1,\ldots, p_n;\sigma)$-continuous multilinear operators.